Two decades ago a construction of correlators of rational two-dimensional conformal field theories as elements in spaces of conformal blocks has been accomplished. An inconvenient limiting feature of that construction is that it heavily relies on three-dimensional geometry.
I describe a purely two-dimensional construction of RCFT correlators that instead makes use of string nets colored by the modular fusion category C of representations of the chiral algebra of the CFT. The construction applies to arbitrary world sheets, including physical boundaries, topological defects and any type of field insertions. It has the potential to extend beyond the realm of rational CFTs.
A bicategorical variant of string nets, based on the bicategory Fr(C) of Frobenius algebras in C rather than on C, allows one to capture neatly the local rules for defects which relate world sheets sharing the same correlator and to give an adequate definition of the mapping class group for surfaces with defects.